Numerical solution of a Cauchy problem for the Laplace equation

We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.